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Stationary max-stable fields associated to negative definite functions
Kabluchko, Zakhar
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Schlather, Martin
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Haan, Laurens de
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Document Type:
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Article
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Year of publication:
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2009
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The title of a journal, publication series:
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The Annals of Probability
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Volume:
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37
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Issue number:
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5
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Page range:
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2042-2065
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Place of publication:
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New York, NY [u.a.]
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Publishing house:
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Inst. of Mathematical Statistics
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ISSN:
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0091-1798 , 2168-894X
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Publication language:
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English
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Institution:
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School of Business Informatics and Mathematics > Applied Stochastics (Schlather 2012-)
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Subject:
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510 Mathematics
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Keywords (English):
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Stationary max-stable processes, Gaussian processes, Poisson point processes, extremes
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Abstract:
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Let Wi, i∈ℕ, be independent copies of a zero-mean Gaussian process {W(t), t∈ℝd} with stationary increments and variance σ2(t). Independently of Wi, let ∑i=1∞δUi be a Poisson point process on the real line with intensity e−y dy. We show that the law of the random family of functions {Vi(⋅), i∈ℕ}, where Vi(t)=Ui+Wi(t)−σ2(t)/2, is translation invariant. In particular, the process η(t)=⋁i=1∞Vi(t) is a stationary max-stable process with standard Gumbel margins. The process η arises as a limit of a suitably normalized and rescaled pointwise maximum of n i.i.d. stationary Gaussian processes as n→∞ if and only if W is a (nonisotropic) fractional Brownian motion on ℝd. Under suitable conditions on W, the process η has a mixed moving maxima representation.
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 | Dieser Datensatz wurde nicht während einer Tätigkeit an der Universität Mannheim veröffentlicht, dies ist eine Externe Publikation. |
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